We have to explain the "chemical composition" of some cell structures. Here's what I have so far;

Cytoplasm- Water (H20)?

Cell wall- Protein molecules?

Nucleus- Nucleic acids?

Cell membrane- Also protein molecules?

I'm not having much luck with the internet, as it's fairly slow and most of the links I click on aren't helping. I'm not sure if my answers are correct, and would saying "protein molecules" and "nucleic acids" be suitable to say for a chemical composition?

Yeah, like Mcar said any decent textbook will give you more than enough details of those. Heck, even a Wikipedia search with the word "cell" will do.

But anyway, all your answers are too general, some even wrong. I don't know how much details your teacher wants, but consider adding some more information, such as:

Cytoplasm: mostly water, but also any soluble components the cell has inside it, such as ions and glucose, as well as proteins it produces or takes in, including cytokines, hormones and such. Finally, since cytoplasm is everything contained within the plasma membrane, it also includes the cell organelles the cell may have - in eukaryotic cells these are e.g. the Golgi apparatuses, mitochondria, endoplasmic reticulum etc.

Cell wall: this depends whether the given cell is a plant, a fungus or a bacterium - plant cell walls mostly contain cellulose, fungi have chitine and bacteria can have e.g. peptidoglygan or glycoproteins.

Nucleus: nucleus contains the nuclear envelope, as well as the nucleic acids inside it. There are also proteins present, some (such as histones) bound to the DNA, some free (e.g. enzymes).

Cell membrane: this is composed of mostly phospholipids, to which many proteins and glycoproteins (e.g. receptors) are anchored.

So, decide what level of depth your answer requires, pick your favourite cell biology text book or put the forementioned topics into Wikipedia search box and just look at the composition of those structures. If your teacher wants the precise chemical formula of the molecules involved, your textbook and Wikipedia will give you those as well. It should be pretty straightforward, but if you still have problems, let us know.